demo.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "07aba147-c9d6-406a-afc5-67b66595d094",
   "metadata": {},
   "source": [
    "Almost everything in the inference study can be considered as doing hypothesis tests. And in most of the cases, any hypothesis $H\\in\\mathcal{H}$ can be described as the subset of the full parameter space. \n",
    "\n",
    "When doing a profiled likelihood test, we factorize the full parameter space as $\\Theta\\times\\Phi$, where $\\Theta$ is the space of interesting parameters and $\\Phi$ is the space of nuisance parameters which will be always profiled out. Rigorously, for a given interesting parameter $\\theta\\in\\Theta$, the corresponding hypothesis should be written as $H_{\\{\\theta\\}\\times\\Phi}$ but I will usually be lazy to just call it $H_\\theta$. And the test statistic we usually use is defined as \n",
    "\n",
    "$$t(X|\\theta) = -2\\log\\frac{L(X|\\theta,\\hat{\\hat{\\phi}})}{L(X|\\hat{\\theta}, \\hat{\\phi})}$$\n",
    "\n",
    "where $\\hat{\\theta}$, $\\hat{\\phi}$ maximize the likelihood $L$ given observation $X$, and $\\hat{\\hat{\\phi}}(\\theta) = \\arg\\max_{\\phi\\in\\Phi} L(X|\\theta, \\phi)$. And Wilk's theorem shows that when the sample size is large, $t$ obeys a chi-square distribution asymptotically with degrees of freedom $\\dim{\\Theta} $.\n",
    "\n",
    "However this asymptotic condition does not always hold. Following Neyman's construction, the key problem is \"if $\\theta, \\phi$ are the true parameters, what is the distribution of $t$\", practically meaning that given $\\theta, \\phi$ we should simulate $X$ from $H_{\\theta,\\phi}$ many times and calculate $t(X|\\theta)$ to get the distribution. To be clear, only two functions are the keys in the package:\n",
    "\n",
    "  1. simulate $X$ from $H_{\\theta,\\phi}$, done by `likelihood.simulate(param)`\n",
    "  2. calculate $t(X|\\theta)$, done by `likelihood.chi2(param, param_profiled_guess)`\n",
    "\n",
    "Now I will show a simple example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "27217b31-f7eb-486d-a977-607d1511dab6",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import inference as inf"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2c6ad4b-09a4-4417-93ef-194494af29e2",
   "metadata": {},
   "source": [
    "First before anything, we set the `url_base` to be the prefix of the path of any used templates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "982b9305-f3dd-4868-913b-15a6c97612d7",
   "metadata": {},
   "outputs": [],
   "source": [
    "inf.set_global_config({'url_base': '/home/zihaoxu/template_v14_SR0/'})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f5ac562-557d-4fa3-97c0-082a06a026e1",
   "metadata": {},
   "source": [
    "# Define a likelihood"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23227a7e-102b-427b-9ce7-ae61b9aaae93",
   "metadata": {},
   "source": [
    "In this example, we consider a 50-GeV WIMP search with ER and NR background, and the likelihood is defined by\n",
    "\n",
    "$$\\begin{aligned}\n",
    "L(\\mathrm{data}|\\lambda_{\\mathrm{ER}}, \\lambda_{\\mathrm{NR}}, \\lambda_{\\mathrm{WIMP}}) &= \\mathrm{Poiss}(N|\\lambda_{\\mathrm{tot}})\\times\\prod_{i=0}^N\\left(\\sum_{s\\in\\{\\mathrm{ER, NR, WIMP}\\}}\\frac{\\lambda_s}{\\lambda_{\\mathrm{tot}}}f_{s,i}\\right)\\times\\mathrm{Gauss}(\\lambda_{\\mathrm{NR}}|\\mu_{\\mathrm{NR}}, \\sigma_{\\mathrm{NR}})\n",
    "\\end{aligned}$$\n",
    "\n",
    "The definition is already done in `aptinf.demo.testLikelihood` and can be viewed by `likelihood.view()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e630e8f7-d649-4d87-bf2c-e2a0dd3f06d9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Poisson(\n",
      "\tn = len(data)\n",
      "\tlam = 10**lg_nr_rate + 10**lg_er_rate + wimp.norm * 10**lg_sig_mul\n",
      ")\n",
      "MultiSourceUnbinnedPDF(\n",
      "\tevents = data\n",
      "\ttemplates = (nr, er, wimp)\n",
      "\tweights = (10**lg_nr_rate, 10**lg_er_rate, wimp.norm * 10**lg_sig_mul)\n",
      ")\n",
      "Gaussian(\n",
      "\tx = 10**lg_nr_rate\n",
      "\tmu = nr.norm\n",
      "\tstd = nr.norm * nr_rate_relative_std\n",
      ")\n"
     ]
    }
   ],
   "source": [
    "likelihood = inf.demo.testLikelihood()\n",
    "likelihood.view()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb324506-c21e-4b27-875f-1b8fdfe4de5f",
   "metadata": {},
   "source": [
    "Every time we want to calculate the likelihood, we have to feed the following things. Here `'data'` is also in the `likelihood.param_needed` due to some technical problem but don't confuse yourself."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "82562eba-b956-48f4-90f1-27697d4bf439",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'data', 'lg_er_rate', 'lg_nr_rate', 'lg_sig_mul'}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "likelihood.param_needed"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da73667a-4e76-4372-8dec-e30e72043dad",
   "metadata": {},
   "source": [
    "# Simulate data"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0607a820-6c09-41df-84ef-996b4d718cef",
   "metadata": {},
   "source": [
    "To simulate data, we need to input all parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "139a7b4f-4f6d-440c-9016-33b1c907fd12",
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood.set_data_from_toymc({'lg_er_rate': 2,\n",
    "                                'lg_nr_rate': np.log10(likelihood.nr.norm),\n",
    "                                'lg_sig_mul': -10})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eea55864-a09b-43e9-ab23-ec0b650ce4fd",
   "metadata": {},
   "source": [
    "Then we calculate the max log likelihood. It's stored inside the class since it's slow to compute could be used many times. We suggest to be careful with the initial guess. If the likelihood is very flat near the initial guess, then the optimizer might NOT give the correct function value."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "cd12378f-2084-442a-9e17-1981830c6d86",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'lg_er_rate': 1.9664847058736348,\n",
       " 'lg_nr_rate': -0.3214876392770275,\n",
       " 'lg_sig_mul': -7.073713030442119}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "likelihood.set_max_loglikelihood(param_guess={'lg_er_rate': 2,\n",
    "                                              'lg_nr_rate': np.log10(likelihood.nr.norm),\n",
    "                                              'lg_sig_mul': 0})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd77db49-bb68-44a1-97b7-5267b7c51536",
   "metadata": {},
   "source": [
    "# Calculate $t$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1b3de92-24a0-47d6-8514-a27f0dac568d",
   "metadata": {},
   "source": [
    "The profiled log likelihood ratio is called by `likelihood.chi2(param, param_profiled_guess)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "13070be6-64de-441a-a0d1-1bd049b78b2f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def t(lg_sig_mul):\n",
    "    return likelihood.chi2({'lg_sig_mul': lg_sig_mul},\n",
    "                           {'lg_er_rate': 2, 'lg_nr_rate': 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "cdc7f521-4a44-400b-8be5-9f4bc3fe8071",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-4, -2, 100)\n",
    "y = [t(xx) for xx in x]\n",
    "\n",
    "plt.plot(10**x, y)\n",
    "plt.xscale('log')\n",
    "plt.xlim(1e-4, 1e-2)\n",
    "plt.ylim(0, None)\n",
    "plt.xlabel('WIMP rate multiplier')\n",
    "plt.ylabel('Profiled log likelihood ratio')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d2c0033b-08e6-4187-983e-9bc498f29c00",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:XENONnT_development]",
   "language": "python",
   "name": "conda-env-XENONnT_development-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.16"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}